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The Radiance Equation

The Radiance Equation. Mel Slater. Outline. Introduction Light Simplifying Assumptions Radiance Reflectance The Radiance Equation Traditional Rendering Solutions Visibility Conclusions. Introduction. Lighting is the central problem of real-time graphics rendering

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The Radiance Equation

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  1. The Radiance Equation Mel Slater

  2. Outline • Introduction • Light • Simplifying Assumptions • Radiance • Reflectance • The Radiance Equation • Traditional Rendering Solutions • Visibility • Conclusions

  3. Introduction • Lighting is the central problem of real-time graphics rendering • Arbitrary shaped lights • Changes in lighting conditions • Real-time shadows • Real-time reflections • Mixtures of many different types of surface

  4. Introduction • Real-time walkthrough with global illumination • Possible under limited conditions • Radiosity (diffuse surfaces only) • Real-time interaction • Not possible except for special case local illumination • Why is the problem so hard?

  5. Light • Visible light is electromagnetic radiation with wavelengths approximately in the range from 400nm to 700nm 400nm 700nm

  6. Light: Photons • Light can be viewed as wave or particle phenomenon • Particles are photons • packets of energy which travel in a straight line in vaccuum with velocity c (300,000m.p.s.) • The problem of how light interacts with surfaces in a volume of space is an example of a transport problem.

  7. Light: Radiant Power •  denotes the radiant energy or flux in a volume V. • The flux is the rate of energy flowing through a surface per unit time (watts). • The energy is proportional to the particle flow, since each photon carries energy. • The flux may be thought of as the flow of photons per unit time.

  8. Light: Flux Equilibrium • Total flux in a volume in dynamic equilibrium • Particles are flowing • Distribution is constant • Conservation of energy • Total energy input into the volume = total energy that is output by or absorbed by matter within the volume.

  9. Light: Fundamental Equation • Input • Emission – emitted from within volume • Inscattering – flows from outside • Output • Streaming – without interaction with matter in the volume • Outscattering – reflected out from matter • Absorption – by matter within the volume • Input = Output

  10. Light: Equation • (p,) denotes flux at pV, in direction  • It is possible to write down an integral equation for (p,) based on: • Emission+Inscattering = Streaming+Outscattering + Absorption • Complete knowledge of (p,) provides a complete solution to the graphics rendering problem. • Rendering is about solving for (p,).

  11. Simplifying Assumptions • Wavelength independence • No interaction between wavelengths (no fluorescence) • Time invariance • Solution remains valid over time unless scene changes (no phosphorescence) • Light transports in a vacuum (non-participating medium) – • ‘free space’ – interaction only occurs at the surfaces of objects

  12. Radiance • Radiance (L) is the flux that leaves a surface, per unit projected area of the surface, per unit solid angle of direction. n d = L dA cos d L  dA

  13. n d L  dA Radiance • For computer graphics the basic particle is not the photon and the energy it carries but the ray and its associated radiance. Radiance is constant along a ray.

  14. Radiance: Radiosity, Irradiance • Radiosity - is the flux per unit area that radiates from a surface, denoted by B. • d = B dA • Irradiance is the flux per unit area that arrives at a surface, denoted by E. • d = E dA

  15. Radiosity and Irradiance • L(p,) is radiance at p in direction  • E(p,) is irradiance at p in direction  • E(p,) = (d/dA) = L(p,) cos d

  16. f(p, i ,r ) i r Reflected ray Incident ray Illumination hemisphere Reflectance • BRDF • Bi-directional • Reflectance • Distribution • Function • Relates • Reflected radiance to incoming irradiance

  17. Reflectance: BRDF • Reflected Radiance = BRDFIrradiance • L(p, r ) = f(p,i ,r ) E(p, i ) • = f(p,i ,r ) L(p, i ) cosi di • In practice BRDF’s hard to specify • Rely on ideal types • Perfectly diffuse reflection • Perfectly specular reflection • Glossy reflection • BRDFs taken as additive mixture of these

  18. The Radiance Equation • Radiance L(p,  ) at a point p in direction  is the sum of • Emitted radiance Le(p,  ) • Total reflected radiance Radiance = Emitted Radiance + Total Reflected Radiance

  19. The Radiance Equation: Reflection • Total reflected radiance in direction : • f(p,i ,) L(p, i ) cosi di • Radiance Equation: • L(p,  ) = Le(p,  ) +  f(p,i ,) L(p, i ) cosi di • (Integration over the illumination hemisphere)

  20. The Radiance Equation • p is considered to be on a surface, but can be anywhere, since radiance is constant along a ray, trace back until surface is reached at p’, then • L(p, i) = L(p’, i) L(p, ) depends on allL(p*, i) which in turnare recursively defined. p* i L(p,  ) p The radiance equation models global illumination.

  21. Traditional Solutions to the Radiance Equation • The radiance equation embodies totality of all 2D projections (view). • Extraction of a 2D projection to form an image is called rendering.

  22. Traditional Solutions

  23. (Image Based Rendering) • IBR not a ‘traditional’ solution • Images for a new view constructed from a large collection of existing images • No lighting computations at all. • Light Field Rendering specific instance to be discussed later.

  24. Visibililty • Where does an incident ray through the image plane come from? • Which surface? • Ray tracing in principle has to search all surfaces for possible intersections • Radiosity has to include visibility in form-factor calculations between surfaces • Real-time rendering solves visibility problem on a pixel by pixel basis (z-buffer). • Major complication for large scenes • We will see later that LFR does not have this visibility problem.

  25. Conclusions • Graphics rendering is concerned with solution of integral radiance equation • Traditional solutions are various kinds of approximations to this equation. • Rendering is the process of extracting images from the equation. • Rendering may be view dependent or independent, together with a global or local illumination solution.

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